Moment maps, cobordisms, and Hamiltonian group actions:
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| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
[2002]
|
| Series: | Mathematical surveys and monographs
Volume 98 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Inhaltsverzeichnis |
| Physical Description: | viii, 350 Seiten Illustrationen |
| ISBN: | 0821805029 9780821805022 |
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MARC
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| 245 | 1 | 0 | |a Moment maps, cobordisms, and Hamiltonian group actions |c Victor Guillemin, Viktor Ginzburg, Yael Karshon |
| 264 | 1 | |a Providence, Rhode Island |b American Mathematical Society |c [2002] | |
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| 650 | 7 | |a MECÂNICA HAMILTONIANA |2 larpcal | |
| 650 | 4 | |a Symplectic geometry | |
| 650 | 4 | |a Cobordism theory | |
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Record in the Search Index
| _version_ | 1804136163456319488 |
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| adam_text | Contents
Chapter 1. Introduction 1
1. Topological aspects of Hamiltonian group actions 1
2. Hamiltonian cobordism 4
3. The linearization theorem and non compact cobordisms 5
4. Abstract moment maps and non degeneracy 7
5. The quantum linearization theorem and its applications 8
6. Acknowledgements 10
Part 1. Cobordism 13
Chapter 2. Hamiltonian cobordism 15
1. Hamiltonian group actions 15
2. Hamiltonian geometry 21
3. Compact Hamiltonian cobordisms 24
4. Proper Hamiltonian cobordisms 27
5. Hamiltonian complex cobordisms 29
Chapter 3. Abstract moment maps 31
1. Abstract moment maps: definitions and examples 31
2. Proper abstract moment maps 33
3. Cobordism 34
4. First examples of proper cobordisms 37
5. Cobordisms of surfaces 39
6. Cobordisms of linear actions 42
Chapter 4. The linearization theorem 45
1. The simplest case of the linearization theorem 45
2. The Hamiltonian linearization theorem 47
3. The linearization theorem for abstract moment maps 51
4. Linear torus actions 52
5. The right hand side of the linearization theorems 56
6. The Duistermaat Heckman and Guillemin Lerman Sternberg formulas 58
Chapter 5. Reduction and applications 63
1. (Pre )symplectic reduction 63
2. Reduction for abstract moment maps 65
3. The Duistermaat Heckman theorem 69
4. Kahler reduction 72
5. The complex Delzant construction 73
6. Cobordism of reduced spaces 81
V
vi CONTENTS
7. Jeffrey Kirwan localization 82
8. Cutting 84
Part 2. Quantization 87
Chapter 6. Geometric quantization 89
1. Quantization and group actions 89
2. Pre quantization 90
3. Pre quantization of reduced spaces 96
4. Kirillov Kostant pre quantization 99
5. Polarizations, complex structures, and geometric quantization 102
6. Dolbeault Quantization and the Riemann Roch formula 110
7. Stable complex quantization and Spinc quantization 113
8. Geometric quantization as a push forward 117
Chapter 7. The quantum version of the linearization theorem 119
1. The quantization of Cd 119
2. Partition functions 125
3. The character of Q(Cd) 130
4. A quantum version of the linearization theorem 134
Chapter 8. Quantization commutes with reduction 139
1. Quantization and reduction commute 139
2. Quantization of stable complex toric varieties 141
3. Linearization of [Q,R]=0 145
4. Straightening the symplectic and complex structures 149
5. Passing to holomorphic sheaf cohomology 150
6. Computing global sections; the lit set 152
7. The Cech complex 155
8. The higher cohomology 157
9. Singular [Q,R]=0 for non symplectic Hamiltonian G manifolds 159
10. Overview of the literature 162
Part 3. Appendices 165
Appendix A. Signs and normalization conventions 167
1. The representation of G on C°°(M) 167
2. The integral weight lattice 168
3. Connection and curvature for principal torus bundles 169
4. Curvature and Chern classes 171
5. Equivariant curvature; integral equivariant cohomology 172
Appendix B. Proper actions of Lie groups 173
1. Basic definitions 173
2. The slice theorem 178
3. Corollaries of the slice theorem 182
4. The Mostow Palais embedding theorem 189
5. Rigidity of compact group actions 191
Appendix C. Equivariant cohomology 197
1. The definition and basic properties of equivariant cohomology 197
CONTENTS vii
2. Reduction and cohomology 201
3. Additivity and localization 203
4. Formality 205
5. The relation between Hg and HJ 208
6. Equivariant vector bundles and characteristic classes 211
7. The Atiyah Bott Berline Vergne localization formula 217
8. Applications of the Atiyah Bott Berline Vergne localization formula 222
9. Equivariant homology 226
Appendix D. Stable complex and Spinc structures 229
1. Stable complex structures 229
2. Spinc structures 238
3. Spinc structures and stable complex structures 248
Appendix E. Assignments and abstract moment maps 257
1. Existence of abstract moment maps 257
2. Exact moment maps 263
3. Hamiltonian moment maps 265
4. Abstract moment maps on linear spaces are exact 269
5. Formal cobordism of Hamiltonian spaces 273
Appendix F. Assignment cohomology 279
1. Construction of assignment cohomology 279
2. Assignments with other coefficients 281
3. Assignment cohomology for pairs 283
4. Examples of calculations of assignment cohomology 285
5. Generalizations of assignment cohomology 287
Appendix G. Non degenerate abstract moment maps 289
1. Definitions and basic examples 289
2. Global properties of non degenerate abstract moment maps 290
3. Existence of non degenerate two forms 294
Appendix H. Characteristic numbers, non degenerate cobordisms, and
non virtual quantization 301
1. The Hamiltonian cobordism ring and characteristic classes 301
2. Characteristic numbers 304
3. Characteristic numbers as a full system of invariants 305
4. Non degenerate cobordisms 308
5. Geometric quantization 310
Appendix I. The Kawasaki Riemann Roch formula 315
1. Todd classes 315
2. The Equivariant Riemann Roch Theorem 316
3. The Kawasaki Riemann Roch formula I: finite abelian quotients 320
4. The Kawasaki Riemann Roch formula II: torus quotients 323
Appendix J. Cobordism invariance of the index of a transversally elliptic
operator by Maxim Braverman 327
1. The Spinc Dirac operator and the Spinc quantization 327
2. The summary of the results 329
viii CONTENTS
3. Transversally elliptic operators and their indexes 331
4. Index of the operator Bo 333
5. The model operator 335
6. Proof of Theorem 1 336
Bibliography 339
Index 349
|
| adam_txt |
Contents
Chapter 1. Introduction 1
1. Topological aspects of Hamiltonian group actions 1
2. Hamiltonian cobordism 4
3. The linearization theorem and non compact cobordisms 5
4. Abstract moment maps and non degeneracy 7
5. The quantum linearization theorem and its applications 8
6. Acknowledgements 10
Part 1. Cobordism 13
Chapter 2. Hamiltonian cobordism 15
1. Hamiltonian group actions 15
2. Hamiltonian geometry 21
3. Compact Hamiltonian cobordisms 24
4. Proper Hamiltonian cobordisms 27
5. Hamiltonian complex cobordisms 29
Chapter 3. Abstract moment maps 31
1. Abstract moment maps: definitions and examples 31
2. Proper abstract moment maps 33
3. Cobordism 34
4. First examples of proper cobordisms 37
5. Cobordisms of surfaces 39
6. Cobordisms of linear actions 42
Chapter 4. The linearization theorem 45
1. The simplest case of the linearization theorem 45
2. The Hamiltonian linearization theorem 47
3. The linearization theorem for abstract moment maps 51
4. Linear torus actions 52
5. The right hand side of the linearization theorems 56
6. The Duistermaat Heckman and Guillemin Lerman Sternberg formulas 58
Chapter 5. Reduction and applications 63
1. (Pre )symplectic reduction 63
2. Reduction for abstract moment maps 65
3. The Duistermaat Heckman theorem 69
4. Kahler reduction 72
5. The complex Delzant construction 73
6. Cobordism of reduced spaces 81
V
vi CONTENTS
7. Jeffrey Kirwan localization 82
8. Cutting 84
Part 2. Quantization 87
Chapter 6. Geometric quantization 89
1. Quantization and group actions 89
2. Pre quantization 90
3. Pre quantization of reduced spaces 96
4. Kirillov Kostant pre quantization 99
5. Polarizations, complex structures, and geometric quantization 102
6. Dolbeault Quantization and the Riemann Roch formula 110
7. Stable complex quantization and Spinc quantization 113
8. Geometric quantization as a push forward 117
Chapter 7. The quantum version of the linearization theorem 119
1. The quantization of Cd 119
2. Partition functions 125
3. The character of Q(Cd) 130
4. A quantum version of the linearization theorem 134
Chapter 8. Quantization commutes with reduction 139
1. Quantization and reduction commute 139
2. Quantization of stable complex toric varieties 141
3. Linearization of [Q,R]=0 145
4. Straightening the symplectic and complex structures 149
5. Passing to holomorphic sheaf cohomology 150
6. Computing global sections; the lit set 152
7. The Cech complex 155
8. The higher cohomology 157
9. Singular [Q,R]=0 for non symplectic Hamiltonian G manifolds 159
10. Overview of the literature 162
Part 3. Appendices 165
Appendix A. Signs and normalization conventions 167
1. The representation of G on C°°(M) 167
2. The integral weight lattice 168
3. Connection and curvature for principal torus bundles 169
4. Curvature and Chern classes 171
5. Equivariant curvature; integral equivariant cohomology 172
Appendix B. Proper actions of Lie groups 173
1. Basic definitions 173
2. The slice theorem 178
3. Corollaries of the slice theorem 182
4. The Mostow Palais embedding theorem 189
5. Rigidity of compact group actions 191
Appendix C. Equivariant cohomology 197
1. The definition and basic properties of equivariant cohomology 197
CONTENTS vii
2. Reduction and cohomology 201
3. Additivity and localization 203
4. Formality 205
5. The relation between Hg and HJ 208
6. Equivariant vector bundles and characteristic classes 211
7. The Atiyah Bott Berline Vergne localization formula 217
8. Applications of the Atiyah Bott Berline Vergne localization formula 222
9. Equivariant homology 226
Appendix D. Stable complex and Spinc structures 229
1. Stable complex structures 229
2. Spinc structures 238
3. Spinc structures and stable complex structures 248
Appendix E. Assignments and abstract moment maps 257
1. Existence of abstract moment maps 257
2. Exact moment maps 263
3. Hamiltonian moment maps 265
4. Abstract moment maps on linear spaces are exact 269
5. Formal cobordism of Hamiltonian spaces 273
Appendix F. Assignment cohomology 279
1. Construction of assignment cohomology 279
2. Assignments with other coefficients 281
3. Assignment cohomology for pairs 283
4. Examples of calculations of assignment cohomology 285
5. Generalizations of assignment cohomology 287
Appendix G. Non degenerate abstract moment maps 289
1. Definitions and basic examples 289
2. Global properties of non degenerate abstract moment maps 290
3. Existence of non degenerate two forms 294
Appendix H. Characteristic numbers, non degenerate cobordisms, and
non virtual quantization 301
1. The Hamiltonian cobordism ring and characteristic classes 301
2. Characteristic numbers 304
3. Characteristic numbers as a full system of invariants 305
4. Non degenerate cobordisms 308
5. Geometric quantization 310
Appendix I. The Kawasaki Riemann Roch formula 315
1. Todd classes 315
2. The Equivariant Riemann Roch Theorem 316
3. The Kawasaki Riemann Roch formula I: finite abelian quotients 320
4. The Kawasaki Riemann Roch formula II: torus quotients 323
Appendix J. Cobordism invariance of the index of a transversally elliptic
operator by Maxim Braverman 327
1. The Spinc Dirac operator and the Spinc quantization 327
2. The summary of the results 329
viii CONTENTS
3. Transversally elliptic operators and their indexes 331
4. Index of the operator Bo 333
5. The model operator 335
6. Proof of Theorem 1 336
Bibliography 339
Index 349 |
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| id | DE-604.BV022190806 |
| illustrated | Illustrated |
| index_date | 2024-07-02T16:21:09Z |
| indexdate | 2024-07-09T20:52:02Z |
| institution | BVB |
| isbn | 0821805029 9780821805022 |
| language | English |
| lccn | 2002074590 |
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| physical | viii, 350 Seiten Illustrationen |
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| series2 | Mathematical surveys and monographs |
| spelling | Guillemin, Victor 1937- Verfasser (DE-588)12110172X aut Moment maps, cobordisms, and Hamiltonian group actions Victor Guillemin, Viktor Ginzburg, Yael Karshon Providence, Rhode Island American Mathematical Society [2002] © 2002 viii, 350 Seiten Illustrationen txt rdacontent n rdamedia nc rdacarrier Mathematical surveys and monographs Volume 98 GEOMETRIA DIFERENCIAL larpcal MECÂNICA HAMILTONIANA larpcal Symplectic geometry Cobordism theory Geometric quantization Symplektische Geometrie (DE-588)4194232-2 gnd rswk-swf Kobordismus (DE-588)4148171-9 gnd rswk-swf Geometrische Quantisierung (DE-588)4156720-1 gnd rswk-swf Symplektische Geometrie (DE-588)4194232-2 s Kobordismus (DE-588)4148171-9 s Geometrische Quantisierung (DE-588)4156720-1 s DE-604 Ginzburg, Victor 1957- Verfasser (DE-588)115025103 aut Karshon, Yael 1964- Verfasser (DE-588)1146225482 aut Erscheint auch als Online-Ausgabe 978-1-4704-1325-5 Mathematical surveys and monographs Volume 98 (DE-604)BV000018014 98 http://www.loc.gov/catdir/toc/fy033/2002074590.html Inhaltsverzeichnis HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015402375&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Guillemin, Victor 1937- Ginzburg, Victor 1957- Karshon, Yael 1964- Moment maps, cobordisms, and Hamiltonian group actions Mathematical surveys and monographs GEOMETRIA DIFERENCIAL larpcal MECÂNICA HAMILTONIANA larpcal Symplectic geometry Cobordism theory Geometric quantization Symplektische Geometrie (DE-588)4194232-2 gnd Kobordismus (DE-588)4148171-9 gnd Geometrische Quantisierung (DE-588)4156720-1 gnd |
| subject_GND | (DE-588)4194232-2 (DE-588)4148171-9 (DE-588)4156720-1 |
| title | Moment maps, cobordisms, and Hamiltonian group actions |
| title_auth | Moment maps, cobordisms, and Hamiltonian group actions |
| title_exact_search | Moment maps, cobordisms, and Hamiltonian group actions |
| title_exact_search_txtP | Moment maps, cobordisms, and Hamiltonian group actions |
| title_full | Moment maps, cobordisms, and Hamiltonian group actions Victor Guillemin, Viktor Ginzburg, Yael Karshon |
| title_fullStr | Moment maps, cobordisms, and Hamiltonian group actions Victor Guillemin, Viktor Ginzburg, Yael Karshon |
| title_full_unstemmed | Moment maps, cobordisms, and Hamiltonian group actions Victor Guillemin, Viktor Ginzburg, Yael Karshon |
| title_short | Moment maps, cobordisms, and Hamiltonian group actions |
| title_sort | moment maps cobordisms and hamiltonian group actions |
| topic | GEOMETRIA DIFERENCIAL larpcal MECÂNICA HAMILTONIANA larpcal Symplectic geometry Cobordism theory Geometric quantization Symplektische Geometrie (DE-588)4194232-2 gnd Kobordismus (DE-588)4148171-9 gnd Geometrische Quantisierung (DE-588)4156720-1 gnd |
| topic_facet | GEOMETRIA DIFERENCIAL MECÂNICA HAMILTONIANA Symplectic geometry Cobordism theory Geometric quantization Symplektische Geometrie Kobordismus Geometrische Quantisierung |
| url | http://www.loc.gov/catdir/toc/fy033/2002074590.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015402375&sequence=000004&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000018014 |
| work_keys_str_mv | AT guilleminvictor momentmapscobordismsandhamiltoniangroupactions AT ginzburgvictor momentmapscobordismsandhamiltoniangroupactions AT karshonyael momentmapscobordismsandhamiltoniangroupactions |
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